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Two-Part Tariffs versus Linear Pricing BetweenManufacturers and Retailers : Empirical Tests on
Differentiated Products Markets
Céline Bonnet∗, Pierre Dubois†, Michel Simioni‡
First Version : June 2004. This Version : October 2004§
Résumé
We present a methodology allowing to introduce manufacturers and retailers ver-tical contracting in their pricing strategies on a differentiated product market. Weconsider in particular two types of non linear pricing relationships, one where resaleprice maintenance is used with two part tariffs contracts and one where no resale pricemaintenance is allowed in two part tariffs contracts. Our contribution allows to recoverprice-cost margins at the manufacturer and retailer levels from estimates of demandparameters. The methodology allows then to test between different hypothesis on thecontracting and pricing relationships between manufacturers and retailers in the su-permarket industry using exogenous variables supposed to shift the marginal costs ofproduction and distribution. We apply empirically this method to study the marketfor retailing bottled water in France. Our empirical evidence shows that manufactu-rers and retailers use non linear pricing contracts and in particular two part tariffscontracts with resale price maintenance.
Key words : vertical contracts, two part tariffs, double marginalization, collusion,competition, manufacturers, retailers, differentiated products, water, non nested tests.
∗University of Toulouse (INRA, GREMAQ)†University of Toulouse (INRA, IDEI) and CEPR‡University of Toulouse (INRA, IDEI)§We thank Marc Ivaldi, Bruno Jullien, Pascal Lavergne, Thierry Magnac, Vincent Réquillart and Patrick
Rey for useful discussions as well as seminar participants at North Carolina State University and EC2
conference in Marseille. Any remaining errors are ours.
1
1 Introduction
Vertical relationships between manufacturers and retailers seem to be more and more
important in the competition analysis of the supermarket industry and in particular in
food retailing. Issues related to market power on some consumption goods markets ne-
cessarily involve the analysis of competition between producers but also between retailers
and the whole structure of the industry. Consumer welfare depends crucially on these
strategic vertical relationships and the competition or collusion degree of manufacturers
and retailers. The aim of this paper is thus to develop a methodology allowing to estimate
alternative structural models where the role of manufacturers and retailers is explicit.
Previous work on these issues generally does not account for the behavior of retailers in
the manufacturers pricing strategies. One of the reasons is that the data used for these
studies are generally demand side data where only retail prices are observed. Information
on wholesale prices and marginal costs of production or distributions are generally difficult
to obtain. Following Rosse (1970), researchers have thus tried to develop methodologies
allowing to estimate price-cost margins that are necessary for market power analysis and
policy simulations, using only data on the demand side, i.e. sales quantities, market shares
and retail prices. The new literature on empirical industrial organization brought new
methods to address this question with the estimation of structural models of competition
on differentiated products markets such as cars, computers, and breakfast cereals (see,
for example, Berry, 1994, Berry, Levinsohn and Pakes, 1995, and Nevo, 1998, 2000, 2001,
Ivaldi and Verboven, 2001). Until recently, most papers in this literature assume that ma-
nufacturers set prices and that retailers act as neutral pass-through intermediaries or that
they were charging exogenous constant margins. The strategic role of retailers has been
emphasized only recently in the empirical economics and marketing literatures. Actually,
Goldberg and Verboven (2001), Mortimer (2004), Sudhir (2001), Berto Villas Boas (2004)
or Villas-Boas and Zhao (2004) introduce retailers’ strategic behavior. For instance, Sudhir
(2001) considers the strategic interactions between manufacturers and a single retailer on
a local market but focuses exclusively on a linear pricing model leading to double margi-
2
nalization. These recent developments that introduce retailers’ strategic behavior consider
mostly cases where competition between producers and/or retailers remains under linear
pricing. Berto Villas-Boas (2004) extends the Sudhir’s framework to multiple retailers and
considers the possibility that vertical contracts between manufacturers and retailers make
pricing strategies depart from double marginalization by setting alternatively wholesale
margins or retail margins to zero. Using recent theoretical developments due to Rey and
Vergé (2004) characterizing pricing equilibria in the case of competition under non linear
pricing between manufacturers and retailers (namely two part tariffs with or without resale
price maintenance), we extend the analysis taking into account vertical contracts between
manufacturers and retailers.
We present how to test across different hypothesis on the strategic relationships bet-
ween manufacturers and retailers in the supermarket industry and in particular how one
can test whether manufacturers use two part tariffs contracts with retailers. We consider
several alternative models of competition and exchange between manufacturers and retai-
lers on a differentiated product market and test between these alternatives. In particular,
we consider two types of non linear pricing relationships, one where resale price mainte-
nance is used with two part tariffs contracts and one where no resale price maintenance
is allowed in two part tariffs (Rey and Vergé, 2004). Modelling explicitly optimal two
part tariffs contracts (with or without resale price maintenance) allows to recover the pri-
cing strategy of manufacturers and retailers and thus the different price-cost margins. The
price-cost margins are expressed as functions of demand parameters. The estimated values
of these demand parameters allow to recover price-cost margins at the manufacturer and
retailer level using only data on the demand side, i.e. without observing wholesale prices.
Using non nested test procedures, we then show how to test between the different models
using exogenous variables that are supposed to shift the marginal costs of production and
distribution.
We apply this methodology to study the market for retailing bottled water in France.
The paper thus presents the first formal empirical tests of whether or not manufacturers
3
use non linear contracts, and, in particular, two-part tariffs contracts with retailers. Our
empirical evidence shows that, in the French bottled water market, manufacturers and
retailers use non linear pricing contracts and in particular two part tariffs contracts with
resale price maintenance.
In section 2, we first present some stylized facts on the market for bottled water
in France, an industry where the questions of vertical relationships and competition of
manufacturers and retailers seem worth studying. Section 3 presents the main methodo-
logical contribution on the supply side. We show how price-cost margins can be recovered
with demand parameters, in particular when taking explicitly into account two part ta-
riffs contracts. Section 4 presents the demand model, its identification and the estimation
method proposed as well as the testing method between the different models. Section 5
presents the empirical results. A conclusion with future research directions is proposed in
section 6, and some appendices follow.
2 Stylized Facts on the Market for Bottled Water in France
The French market for bottled water is one of the more dynamic sector of the French
food processing industry : the total production of bottled water has increased by 4%
in 2000, and its turnover by 8%. Some 85% of French consumers drink bottled water,
and over two thirds of French bottled water drinkers drink it more than once a day, a
proportion exceeded only in Germany. The French bottled water sector is a highly concen-
trated sector, the first three main manufacturers (Nestlé Waters, Danone, and Castel)
sharing 90% of the total production of the sector. Moreover, given the rarity of natural
springs, entry both for mineral or spring water is rather difficult in this market where there
exist some natural capacity constraints. Compte, Jenny and Rey (2002) comment on the
Nestlé/Perrier Merger case that took place in the beginning of 90’s in Europe and point
out that these capacity constraint are a factor of collusion by themselves in addition to
the high concentration of the sector. This sector can be divided in two major segments :
mineral water and spring water. Natural mineral water benefits some properties favorable
to health, which are officially recognized. Composition must be guaranteed as well as the
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consistency of a set of qualitative criteria : mineral content, visual aspects, and taste. The
mineral water can be marketed if it receives an agreement from the French Ministry of
Health. The exploitation of a spring water source requires only a license provided by local
authorities (Prefectures) and a favorable opinion of the local health committee. Moreover,
the water composition is not required to be constant. The differences between the quality
requirements involved in the certification of the two kinds of bottled water may explain
part of the large difference that exists between the shelf prices of the national mineral wa-
ter brands and the local spring water brands. Moreover, national mineral water brands are
highly advertised. The bottled water products use mainly two kinds of differentiation. The
first kind of differentiation stems from the mineral composition, that is the mineral salts
content, and the second from the brand image conveyed through advertising. Actually,
thanks to data at the aggregate level (Agreste, 1999, 2000, 2002) on food industries and
the bottled water industry, one can remark (see the following Table) that this industry
uses much more advertising than other food industries. Friberg and Ganslandt (2003) also
report a high advertising to revenue ratio for the same industry in Sweden, i.e., 6.8% over
the 1998-2001 period. For comparison, the highest advertising to revenue ratio in the US
food processing industry corresponds to the ready-to-eat breakfast cereal industry and
is of 10.8%. These figures may be interpreted as showing the importance of horizontal
differentiation of products for bottled water.
Year Bottled Water All Food IndustriesPCM Advertising/Revenue PCM Advertising/Revenue
1998 17.38% 12.09% 6.32% 5.57%1999 16.70% 14.91% 6.29% 6.81%2000 13.61% 15.89% 3.40% 8.76%Table : Aggregate Estimates of Margins and Advertising to Sales Ratios.
These aggregate data also allow to compute some accounting price-cost margin1 defined
as value added2 (V A) minus payroll (PR) and advertising expenses (AD) divided by the
value of shipments (TR). As emphasized by Nevo (2001), these accounting estimates can
1The underlying assumptions in the definition of these price-cost margins are that the marginal cost isconstant and is equal to the average variable cost (see Liebowitz, 1982).
2Value added is defined as the value of shipments plus services rendered minus cost of materials, suppliesand containers, fuel, and purchased electrical energy.
5
be considered as an upper bound to the true price-cost margins.
Recently, the degradation of the distribution network of tap water has led to an increase
of bottled water consumption. This increase benefited to the cheapest bottled water, that
is to the local spring water. For instance, the total volume of local spring water sold in
2000 reached closely the total volume of mineral water sold the same year. Households buy
bottled water mostly in supermarkets : some 80% of the total sales of bottled water comes
from supermarkets. Moreover, on average, these sales represent 1.7% of the total turnover
of supermarkets, the bottled water shelf being one of the most productive. French bottled
water manufacturers thus deal mainly their brands through retailing chains. These chains
are also highly concentrated, the market share of the first five accounting for 80.7% of total
food product sales. Moreover, these late years, like other processed food products, these
chains have developed private labels to attract consumers. The increase in the number of
private labels tends to be accompanied by a reduction of the market shares of the main
national brands.
We thus face a relatively concentrated market for which the questions of whether or
not producers may exert bargaining power in their strategic relationships with retailers is
important. The study of competition issues and evaluation of markups, which is crucial
for consumer welfare, has then to take into account the possibility that non linear pricing
may be used between manufacturers and retailers. Two part tariffs are typically relatively
simple contracts that may allow manufacturers to benefit from their bargaining position
in selling national brands. Therefore, we study in the next section different alternative
models of strategic relationships between multiple manufacturers and multiple retailers
that are worth considering.
3 Competition and Vertical Relationships Between Manu-facturers and Retailers
Before presenting our demand model, we present now the modelling of the competition
and vertical relationships between manufacturers and retailers. Given the structure of the
bottled water industry and the retail industry in France, we consider several oligopoly
6
models with different vertical relationships that seem to deserve particular attention. More
precisely, we show how each supply model can be solved to obtain an expression for both
the retailer’s and manufacturer’s price-cost margins just as a function of demand side
parameters. Then using estimates of a differentiated products demand model, we will
be able to estimate empirically these price-cost margins and we will show how we can
test between these competing scenarios. A similar methodology has been used already
for double marginalization scenarios considered below by Sudhir (2001) or Brenkers and
Verboven (2004) or Berto Villas-Boas (2004) but none of the papers in this literature
already considered the particular case of competition in two part tariffs as in Rey and
Vergé (2004).
Let’s first introduce the notations. There are J differentiated products defined by the
couple product-retailer corresponding to J 0 national brands and J − J 0 private labels.
There are F manufacturers and R retailers. We denote by Sr the set of products sold by
retailer r and by Ff the set of products produced by firm f . In the following we present
successively the different oligopoly models that we want to study.
3.1 Linear Pricing and Double Marginalization
In this model, except for private labels, the manufacturers set their prices first, and
retailers follow, setting the retail prices given the wholesale prices. For private label pro-
ducts, prices are chosen by the retailer himself who acts as doing both manufacturing and
retailing. Thus, suppose there are R retailers competing in the retail market and that there
are F manufacturers competing in the wholesale market. We consider that competition is
à la Nash-Bertrand. We solve this vertical model by backward induction considering the
retailer’s problem. The profit Πr of retailer r in a given period (we drop the time subscript
t for ease of presentation) is given by
Πr =Xj∈Sr
(pj −wj − cj)sj(p)M
where pj is the retail price of product j sold by retailer r, wj is the wholesale price paid
by retailer r for product j, cj is the retailer’s (constant) marginal cost of product j, sj(p)
is the market share of product j, p is the vector of all products retail prices and M is the
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size of the market. Assuming that a pure-strategy Bertrand-Nash equilibrium in prices
exists and that equilibrium prices are strictly positive, the price of any brand j sold by
retailer r must satisfy the first-order condition
sj +Xk∈Sr
(pk −wk − ck)∂sk∂pj
= 0, for all j ∈ Sr. (1)
Now, we define Ir (of size (J × J)) as the ownership matrix of the retailer r that is
diagonal and whose elements Ir(j, j) are equal to 1 if the retailer r sells products j and
zero otherwise. Let Sp be the market shares response matrix to retailer prices, containing
the first derivatives of all market shares with respect to all retail prices, i.e.
Sp ≡
⎛⎜⎝∂s1∂p1
. . . ∂sJ∂p1
......
∂s1∂pJ
. . . ∂sJ∂pJ
⎞⎟⎠In vector notation, the first order condition (1) implies that the vector γ of retailer r’s
margins, i.e. the retail price p minus the wholesale price w minus the marginal cost of
production c, is3
γ ≡ p−w− c = − (Ir × Sp × Ir)−1 × Ir × s(p) (2)
Remark that for private label products, this price-cost margin is in fact the total price
cost margin p−µ−c which amounts to replace the wholesale price w by the marginal costof production µ in this formula.
Concerning the manufacturers’ behavior, we also assume that each manufacturer maxi-
mizes his profit choosing the wholesale prices wj of the product j he sells and given the
retailers’ response (1). The profit of manufacturer f is given by
Πf =Xj∈Ff
(wj − µj)sj(p(w))M
where µj is the manufacturer’s (constant) marginal cost of production of product j. As-
suming the existence of a pure strategy Bertrand-Nash equilibrium in wholesale prices
between manufacturers, the first order conditions are
sj +Xk∈Ff
Xl=1,..,J
(wk − µk)∂sk∂pl
∂pl∂wj
= 0, for all j ∈ Ff . (3)
3Remark that in all the following, when we use the inverse of non invertible matrices, it means that we
consider the matrix of generalized inverse which means that for example∙2 00 0
¸−1=
∙1/2 00 0
¸.
8
Consider If the ownership matrix of manufacturer f that is diagonal and whose element
If (j, j) is equal to one if j is produced by the manufacturer f and zero otherwise. We
introduce Pw the (J × J) matrix of retail prices responses to wholesale prices, containing
the first derivatives of the J retail prices p with respect to the J 0 wholesale prices w.
Pw ≡
⎛⎜⎜⎜⎜⎜⎜⎝
∂p1∂w1
.. ∂pJ∂wJ0
.. ∂pJ∂w1
......
...∂p1∂wJ0 ..
∂pJ0∂wJ0
.. ∂pJ∂wJ0
0 .. 0 .. 00 .. 0 .. 0
⎞⎟⎟⎟⎟⎟⎟⎠Remark that the last J−J 0 lines of this matrix are zero because they correspond to privatelabels products for which wholesale prices have no meaning.
Then, we can write the first order conditions (3) in matrix form and the vector of manu-
facturer’s margins is4
Γ ≡ w − µ = −(If × Pw × Sp × If )−1 × If × s(p) (4)
The first derivatives of retail prices with respect to wholesale prices depend on the strategic
interactions between manufacturers and retailers. Let’s assume that the manufacturers set
the wholesale prices and retailers follow, setting the retail prices given the wholesale prices.
Therefore, Pw can be deduced from the differentiation of the retailer’s first order conditions
(1) with respect to wholesale price, i.e. for j ∈ Sr and k = 1, .., J 0
Xl=1,..,J
∂sj(p)
∂pl
∂pl∂wj−∂sk(p)
∂pj+Xl∈Sr
∂sl(p)
∂pj
∂pl∂wk
+Xl∈Sr
(pl−wl−cl)X
s=1,..,J
∂2sl(p)
∂pj∂ps
∂ps∂wk
= 0 (5)
Defining Spjp the (J × J)matrix of the second derivatives of the market shares with respect
to retail prices whose element (l, k) is ∂2sk∂pj∂pl
, i.e.
Spjp ≡
⎛⎜⎜⎝∂2s1∂p1∂pj
. . . ∂2sJ∂p1∂pj
... ....
∂2s1∂pJ∂pj
. . . ∂2sJ∂pJ∂pj
⎞⎟⎟⎠We can write equation (5) in matrix form5 :
Pw = Ir × Sp ×£Sp × Ir + Ir × S0p × Ir + (S
p1p × Ir × γ|...|SpJ
p × Ir × γ)× Ir¤−1 (6)
4Rows of this vector that correspond to private labels are zero.5We use the notation (a|b) for horizontal concatenation for vectors (or matrices) a and b.
9
where γ = p−w−c. Equation (6) shows that one can express the manufacturer’s price cost
margins vector Γ = w − µ as depending on the function s(p) by replacing the expression
(6) for Pw in (4).
The expression (6) comes from the assumption that manufacturers act as Stackelberg
leaders in the vertical relationships with retailers. In the case where we would assume
that retailers and manufacturers set simultaneously their prices, we assume like Sudhir
(2001) that only the direct effect of wholesale price on retail price matter through. Thus,
the retailer input cost is accounted for in the retailer’s choice of margin. In this case, the
matrix Pw has to be equal to the following diagonal matrix⎛⎜⎜⎜⎜⎜⎜⎜⎝
1 0 .. .. 0
0. . . . . . . . .
....... . . 1
. . ....
... .. .. 0 00 .. .. 0 0
⎞⎟⎟⎟⎟⎟⎟⎟⎠Then, again one can compute the price-cost margins of the retailer and the manufacturer
under this assumption.
We can also consider the model where retailers and/or manufacturers collude perfectly
just by modifying the ownership matrices. In the case of perfect price collusion between
retailers, instead of using the ownership matrices Ir for each retailers in (2) one can get
the price cost margins of the retail industry by replacing Ir in (2) by the identity matrix
(the situation being equivalent to a retailer in monopoly situation). Similarly, one can get
the price-cost margins vector of manufacturers in the case of perfect collusion by replacing
the ownership matrix If in (4) by a diagonal matrix where diagonal elements are equal to
one except for private labels goods.
Finally, one can also consider the case of a monopoly and we simply need to consider
the joint maximization of profits of all retailers and manufacturers which amounts to
maximize Xj=1,...,J
(pj − µj − cj)sj(p)M
Therefore, the price-cost margins of the full vertically and horizontally integrated structure
10
can be expressed as (retail price minus both marginal costs denoted γ + Γ)
γ + Γ = p− µ− c = −S−1p × s(p) (7)
3.2 Two-Part Tariffs
We now consider the case where manufacturers and retailers can sign two-part tariffs
contracts. We assume that manufacturers have all the bargaining power. To prove the
existence and characterize equilibria in this multiple common agency game is difficult. We
could assume the existence of symmetric subgame perfect Nash equilibria but Rey and
Vergé (2004) prove that some equilibrium exists under some assumptions on the game
played. Actually, assume that manufacturers and retailers play the following game. First,
manufacturers simultaneously propose two-part tariffs contracts to each retailer. These
contracts consist in the specification of franchise fees and wholesale prices but also on
retail prices in the case where manufacturers can use resale price maintenance. Thus we
assume that, for each product, manufacturers propose the contractual terms to retailers
and then, retailers simultaneously accept or reject the offers that are public information.
If one offer is rejected, then all contracts are refused6. If all offers have been accepted,
the retailers simultaneously set their retail prices, demands and contracts are satisfied.
Rey and Vergé (2004) showed (in the two manufacturers - two retailers case) that there
exist some equilibria to this (double) common agency game provided some conditions on
elasticities of demand and on the shape of profit functions are satisfied7. Rey and Vergé
(2004) show that it is always a dominant strategy for manufacturers to set retail prices
in their contracting relationship with retailers. Moreover, with resale price maintenance,
the manufacturer can always replicate the retail price that would emerge and the profit
it would earn without resale price maintenance. We also consider the case where resale
price maintenance would not be used by manufacturers because in some contexts, like in
6This assumption is strong but it happens that the characterization of equilibria in the opposite caseis very difficult (see Rey and Vergé, 2004). However, this assumption means that we should observe allmanufacturers trading will all retailers, which is the case for bottled water in France.
7These technical assumptions require that direct price effects dominate in demand elasticities such thatif all prices increase, demand decreases. The empirical estimation of demand will confirm that this is thecase for bottled water in France. Also it has to be that the monopoly profit function of the industry hasto be single peaked as well as manufacturers revenue functions of the wholesale price vector.
11
France, resale price maintenance may be forbidden and manufacturers thus prefer not to
use it.
In the case of these two part tariffs contracts, the profit function of retailer r is :
Πr =Xs∈Sr
[M(ps −ws − cs)ss(p)− Fs] (8)
where Fs is the franchise paid by the retailer for selling product s.
Manufacturers set their wholesale prices to wk and the franchise fees Fk and choose
the retail’s prices in order to maximize profits which is for firm f equal to
Πf =Xk∈Ff
[M(wk − µk)sk(p) + Fk] (9)
subject to the retailers’ participation constraints Πr ≥ 0, for all r = 1, .., R.
Since the participation constraints are clearly binding (Rey and Vergé, 2004) and ma-
nufacturers choose the fixed fees Fk given the ones of the other manufacturers, one can
replace the expressions of the franchise fee Fk of the binding participation constraint (8)
into the manufacturer’s profit (9) and obtain the following profit for firm f (see details in
appendix 7.1) Xk∈Ff
(pk − µk − ck)sk(p) +Xk 6∈Ff
(pk −wk − ck)sk(p)
Then, the maximization of this objective function depends on whether resale price main-
tenance is used or not by manufacturers.
Two part tariffs with resale price maintenance :
Since manufacturers can capture retail profits through the franchise fees and moreover set
retail prices, the wholesale prices have no direct effect on profit. Rey and Vergé (2004)
showed however that the wholesale prices influence the strategic behavior of competitors.
They show that there exists a continuum of equilibria, one for each wholesale price vector.
For each wholesale price vector w∗, there exists a unique symmetric subgame perfect
equilibrium in which retailers earn zero profit and manufacturers set retail prices to p∗(w∗),
where p∗(w∗) is a decreasing function of w∗ equal to the monopoly price when the wholesale
prices are equal to the marginal cost of production. For our purpose, we choose some
possible equilibria among this multiplicity of equilibria. For a given equilibrium p∗(w∗),
12
the program of manufacturer f is now
max{pk}∈Ff
Xk∈Ff
(pk − µk − ck)sk(p) +Xk 6∈Ff
(p∗k −w∗k − ck)sk(p)
Thus, we can write the first order conditions for this program as
Xk∈Ff
(pk − µk − ck)∂sk(p)
∂pj+ sj(p) +
Xk 6∈Ff
(p∗k −w∗k − ck)∂sk(p)
∂pj= 0 for all j ∈ Ff (10)
Then, depending on the wholesale prices, several cases can be considered. We will consi-
der two cases of interest : first when wholesale prices are equal to the marginal cost of
production (w∗k = µk), second, when wholesale prices are such that the retailer’s price cost
margins are zero (p∗k(w∗k)−w∗k − ck = 0).
First, when w∗k = µk, the first order condition (10) writes
Xk∈Ff
(pk − µk − ck)∂sk(p)
∂pj+ sj(p) +
Xk 6∈Ff
(p∗k − µk − ck)∂sk(p)
∂pj= 0 for all j ∈ Ff
i.e. Xk
(pk − µk − ck)∂sk(p)
∂pj+ sj(p) = 0 for all j ∈ Ff
which gives in matrix notation for manufacturer f
γf + Γf = (p− µ− c) = −(If × Sp)−1If × s(p) (11)
Second, when wholesale prices w∗k are such that p∗k(w
∗k)−w∗k − ck = 0, then (10) becomes
Xk∈Ff
(pk − µk − ck)∂sk(p)
∂pj+ sj(p) = 0 for all j ∈ Ff
In matrix notations, we get for all f = 1, .., F
γf + Γf = (p−w − c) = (p− µ− c) = −(If × Sp × If )−1 × If × s(p)
However, among the continuum of possible equilibria, Rey and Vergé (2004) showed
that the case where wholesale prices are equal to the marginal costs of production is the
equilibrium that would be selected if retailers can provide a retailing effort that increases
demand. Actually, in this case it is worth for the manufacturer to make the retailer residual
claimant of his retailing effort which leads to select this equilibrium wholesale price.
13
In the case of two part tariffs contracts withRPM between manufacturers and retailers,
we assume that the profit maximizing strategic pricing of private label products by retailers
is taken into account by manufacturers when they choose fixed fees and retail prices of
their own products in the contract. This implies that the prices of private label products
chosen by retailers is such that they maximize their profit on these private labels and the
total price cost margin eγr + eΓr for these private labels will be such thateγr + eΓr ≡ p− µ− c = −
³eIr × Sp × eIr´−1 × eIr × s(p) (12)
where eIr is the ownership matrices of private label products of retailer r.Two part tariffs without resale price maintenance :
Let’s consider now that resale price maintenance cannot be used by manufacturers. Since
they cannot choose retail prices, they only set wholesale prices in the following maximiza-
tion program
max{wk}∈Ff
Xk∈Ff
(pk − µk − ck)sk(p) +Xk 6∈Ff
(pk −wk − ck)sk(p)
Then the first order conditions are for all i ∈ Ff
Xk
∂pk∂wi
sk(p)+Xk∈Ff
⎡⎣(pk − µk − ck)Xj
∂sk∂pj
∂pj∂wi
⎤⎦+Xk 6∈Ff
⎡⎣(pk −wk − ck)Xj
∂sk∂pj
∂pj∂wi
⎤⎦ = 0which gives in matrix notation
If × Pw × s(p) + If × Pw × Sp × If × (p− µ− c) + If × Pw × Sp × I−f × (p−w − c) = 0
This implies that the total price cost margin γ+Γ = p−µ−c is such that for all j = 1, .., J :
γ + Γ = (If × Pw × Sp × If )−1 × [−If × Pw × s(p)− If × Pw × Sp × I−f × (p−w − c)]
(13)
that allows us to estimate the price-cost margins with demand parameters using (2) to
replace (p−w − c) and (6) for Pw. Remark again that the formula (2) provides directly
the total price-cost margin obtained by each retailer on its private label product.
We are thus able to obtain the several expressions for price-cost margins at the manu-
facturing or retail levels under the different models considered and function of the demand
parameters.
14
4 Differentiated Products Demand
4.1 The Random Utility Demand Model
We now describe our model of differentiated product demand. We use a standard
random utility model. Actually, denoting Vijt the utility for consumer i of buying good j
at period t, we assume that it can be represented by
Vijt = θjt + εijt
= δj + γt − αpjt + ujt + εijt for j = 1, ., J
where θjt is the mean utility of good j at period t and εijt a (mean zero) individual—
product-specific utility term representing the deviation of individual’s preferences from
the mean θjt.
Moreover, we assume that θjt is the sum of a mean utility δj of product j common to all
consumers, a mean utility γt common to all consumers and products at period t (due to
unobserved preference shocks to period t), a product-time specific unobserved utility term
ujt and an income disutility αpjt where pjt is the price of product j at period t.
Consumers may decide not to purchase any of the products. In this case they choose
an outside good for which the mean part of the indirect utility is normalized to 0, so
that Vi0t = εi0t. Remark that the specification used for θjt is such that one could also
consider that the mean utility of the outside good depends also on its time varying price
p0t without changing the identification of the other demand parameters. Actually, adding
−αp0t to the outside good mean utility is equivalent to adding αp0t to all other goods
mean utility, which would amount to replace γt by γt + αp0t.
Then, we model the distribution of the individual-specific utility term εijt according to
the assumptions of a nested logit model (Ben-Akiva, 1973). Actually, we assume that the
bottled water market can be partitioned into G different groups, each sub-group g contai-
ning Jg products (PG
g=1 Jg = J). With an abuse of notation, we will also denote Jg the set
of products belonging to the sub-group g. Since products belonging to the same subgroup
share a common set of unobserved features, consumers may have correlated preferences
over these features. The distributional assumptions of the nested logit model imply that
15
consumers may have correlated preferences across all products of a given subgroup. In the
bottled water market in France, it seems that customers make a clear difference between
two groups of bottled water (G = 2), mineral and spring water, such that it makes sense to
allow customers to have correlated preferences over such groups (Friberg and Ganslandt
(2003) use a similar demand model).
Assuming that consumers choose one unit of the good that maximizes utility, the
distributional assumptions of the nested logit model yield the following choice probabilities
or market share for each product j, as a function of the price vector pt = (p1t, p2t, ..., pJt) :
sjt(pt) = P
µVijt = max
l=0,1,.,J(Vilt)
¶= sjt/g(pt)× sgt(pt)
where sgt(pt) and sjt/g(pt) denote respectively the probability choice of group g and the
conditional probability of choosing good j conditionally on purchasing a good in group g.
The expressions of these probabilities are given by
sjt/g(pt) =exp
θjt1−σgP
j∈Jg expθjt1−σg
=exp
θjt1−σg
expIgt1−σg
sgt(pt) =
³Pj∈Jg exp
θjt1−σg
´1−σgPG
g=0
³Pj∈Jg exp
θjt1−σg
´1−σg = exp Igtexp It
where Igt, and It, are “inclusive values”, defined by :
Igt = (1− σg) lnXj∈Jg
expθjt
1− σg
It = lnX
g=1,.,G
exp Igt
The parameter σg associated to the subgroup g measures the degree of correlation of
consumer preferences for bottled water belonging to the same subgroup. The conditions
on McFadden’s (1978) Generalized Extreme Value model required for the model to be
consistent with random utility maximization are that σg ∈ [0, 1]. When σg goes to 1,
preferences for products of the same subgroup become perfectly correlated meaning that
these products are perceived as perfect substitutes. When σg goes to 0, preferences for
all products become uncorrelated, and the model reduces to a simple multinomial logit
model. At the aggregate demand level, the parameter σg allows to assess to which extent
16
competition is localized between products from the same subgroup. This specification is
more flexible than a simple multinomial logit specification (since it includes it as a special
case). Actually, in the special case where σg = 0 for g = 1, ..,G, we obtain a simple
multinomial logit model which amounts to assume that εijt is i.i.d. with a type I extreme
value distribution. Then we have
sjt(pt) =exp [θjt]
1 +P
j=1,.,Jexp [θjt]
Removing the time subscript for notational simplicity, the nested logit specification as-
sumption implies that the price-elasticity of demand of good j belonging to group g with
respect to good k is given by
ηjk ≡∂sj∂pk
pksj=
⎧⎪⎨⎪⎩α
1−σg pk[σgsj/g + (1− σg)sj − 1] if j = k and {j, k} ∈ gα
1−σg pk[σgsk/g + (1− σg)sk] if j 6= k and {j, k} ∈ g
αpksk if j ∈ g and k ∈ g0 and g 6= g0
The nested logit model can be interpreted as a special case of the random coefficients logit
models estimated by Berry, Levinsohn and Pakes (1995), Nevo (2001), Petrin (2002) and
others. McFadden and Train (2000) show that any random utility model can be arbitra-
rily approximated by a random coefficient logit model. The nested logit model introduces
restrictions on the underlying model but it has the advantage to be econometrically trac-
table even if potentially restrictive (Berry, 1994, and Berry and Pakes, 2001). However,
one can then test whether the structural restrictions imposed by the nested logit model
are empirically accepted or not.
4.2 Identification and Estimation of the Econometric Model
Following Berry (1994) and Verboven (1996), the random utility model introduced in
the previous section leads to the following equations on the aggregate market shares of
goods j at time t
ln sjt − ln s0t = θjt + σg ln sjt|g + ujt
= δj + γt − αpjt + σg ln sjt|g + ujt (14)
where sjt|g is the relative market share of product j at period t in its group g and s0t is
the market share of the outside good at time t. In the particular case of the simple logit
17
model, this equation becomes
ln sjt − ln s0t = δj + γt − αpjt + ujt (15)
Remark that the full set of time fixed effects γt captures preferences for bottled water
relative to the outside good, and can thus be thought of as accounting for macro-economic
fluctuations (like the weather) that affect the decision to buy bottled water8 but also as
accounting for the outside good price variation across periods.
The error term ujt captures the remaining unobserved product valuations varying across
products and time, e.g. due to unobserved variations in advertising. Of course, the usual
problem of endogeneity of price pjt and relative market shares sjt|g has to be handled
correctly in order to identify and estimate the parameters of these models.
Our identification strategy then relies on the use of instrumental variables. Actually,
thanks to the collection of data on wages, oil, diesel, packaging material and plastic prices
over the period of interest, we construct instruments for prices pjt that are interactions
between characteristics of bottled water and these prices (in particular we use the mineral
versus spring water dummy crossed with these prices). The identification then relies on
the fact that these input prices affect the product prices because they are correlated with
input costs but are not correlated with the idiosyncratic unobserved shocks to preferences
ujt. For the simple logit model, this set of instrumental variables is sufficient, but for the
nested logit model, one has also to take into account the endogeneity of the relative (within
group) market shares. For these relative market shares, our strategy relies on the fact that
the contemporaneous correlation between ln sjt|g and unobserved shocks ujt, which is the
source of the endogeneity problem, can be removed with some suitable projection of the
relative market shares on the hyperplane generated by some observed variables.
Actually, the endogeneity of price makes the relative market share endogenous since it is
correlated with price and thus with idiosyncratic preference shock ujt. One can decompose
the relative market share between a predicted part and an unpredicted part using some
8Similarly, in all the regressions they perform, Friberg and Ganslandt (2003) include also a dummy forthe high demand season, i.e. summer.
18
observed covariates. We estimate the following regression
ln sjt|g = πj + β ln sjt−1|g + ξgjt (16)
and we assume that
corr³ξgjt, ln sjt|g
´6= 0 and corr
³ξgjt, ujt
´= 0
Then, we use ξgjt crossed with the dummy variables for mineral or spring water as additional
instrument for ln sjt|g in our main equation (14). This strategy relies on a structural
assumption about the endogeneity problem raised by the relative market shares ln sjt|g
in our main equation (14). Actually, constructing valid instrumental variables for ln sjt|g
using ξgjt means that the endogeneity problem of ln sjt|g comes through the unobserved non
time-varying effects πj and eventually through the lagged value of the relative market share
that are correlated with shocks ujt. As we will see later, this interpretation is consistent
with empirical tests of the validity of these instruments. A detailed description of (16) is
given in appendix 7.2. The instruments ξgjt prove empirically to be well correlated with the
endogenous variables (as shown by first stage regressions) and satisfy the overidentifying
restrictions.
The identification and estimation of these demand models then permits to evaluate
own and cross price elasticities in this differentiated product demand model.
4.3 Testing Between Alternative Models
We now present how to test between the alternative models once we have estimated
the demand model and obtained the different price-cost margins estimates according to
their expressions obtained in the previous section.
Denoting by h the different models considered, for product j at time t under model h,
we denote γhjt the retailer price cost margin and Γhjt the manufacturer price cost margin.
Using Chjt for the sum of the marginal cost of production and distribution (C
hjt = µhjt+chjt)
we can estimate this marginal cost using prices and price cost margins with
Chjt = pjt − Γhjt − γhjt (17)
19
Let’s now assume that these marginal costs are affected by some exogenous shocks Wjt,
we use the following specification
Chjt = pjt − Γhjt − γhjt =
hexp(ωhj +W 0
jtγh)iηhjt
where ωhj is an unknown product specific parameter, Wjt are observable random shock to
the marginal cost of product j at time t and ηhjt is an unobservable random shock to the
cost. Taking logarithms, we get
lnChjt = ωhj +W 0
jtγh + ln ηhjt (18)
Assuming that corr(ln ηhjt,Wjt) = corr(ln ηhjt, ωhj ) = 0, one can identify and estimate
consistently ωhj , γg, and ηhjt.
Now, for any two models h and h0, one would like to test one model against the other,
that is test between
pjt = Γhjt + γhjt +
hexp(ωhj +W 0
jtγh)iηhjt
and
pjt = Γh0jt + γh
0jt +
hexp(ωh
0j +W 0
jtγh0)iηh
0jt
Using non linear least squares
minγh,ω
hj
Qhn(γh, ω
hj ) = min
γh,ωhj
1
n
Xj,t
³ln ηhjt
´2= min
γh,ωhj
1
n
Xj,t
hln³pjt − Γhjt − γhjt
´− ωhj −W 0
jtγh
i2Then, we use non nested tests (Vuong, 1989, and Rivers and Vuong, 2002) to infer which
model h is statistically the best. The tests we use consist in testing models one against
another. The test of Vuong (1989) applies in the context of maximum likelihood estimation
and thus would apply in our case if one assumes log-normality of ηhjt. Rivers and Vuong
(2002) generalized this kind of test to a broad class of estimation methods including
non linear least squares. Moreover, the Vuong (1989) or the Rivers and Vuong (2002)
approaches do not require that either competing model be correctly specified under the
tested null hypothesis. Indeed, other approaches such as Cox’s tests (see, among others,
Smith, 1992) require such an assumption, i.e. that one of the competing model accurately
20
describes the data. This assumption cannot be sustained when dealing with a real data
set like ours.
Taking any two competing models h and h0, the null hypothesis is that the two non
nested models are asymptotically equivalent when
H0 : limn→∞
nQ̄hn(γh, ω
hj )− Q̄h0
n (γh0 , ωh0j )o= 0
where Q̄hn(γh, ω
hj ) (resp. Q̄
h0n (γh0 , ω
h0j )) is the expectation of a lack-of-fit criterionQ
hn(γh, ω
hj )
(i.e. the opposite of a goodness-of-fit criterion) evaluated for model h (resp. h0) at the
pseudo true values of the parameters of this model, denoted by γh, ωhj (resp. γh0 , ω
h0j ). The
first alternative hypothesis is that h is asymptotically better than h0 when
H1 : limn→∞
nQ̄hn(γh, ω
hj )− Q̄h0
n (γh0 , ωh0j )o< 0
Similarly, the second alternative hypothesis is that h0 is asymptotically better than h when
H2 : limn→∞
nQ̄hn(γh, ω
hj )− Q̄h0
n (γh0 , ωh0j )o> 0
The test statistic Tn captures the statistical variation that characterizes the sample values
of the lack-of-fit criterion and is then defined as a suitably normalized difference of the
sample lack-of-fit criteria, i.e.
Tn =
√n
σ̂hh0
n
nQhn(bγh, bωhj )−Qh0
n (bγh0 , bωh0j )owhere Qh
n(bγh, bωhj ) (resp. Qh0n (bγh0 , bωh0j )) is the sample lack-of-fit criterion evaluated for mo-
del h (resp. h0) at the estimated values of the parameters of this model, denoted by bγh, bωhj(resp. bγh0 , bωh0j ). σ̂hh0n denotes the estimated value of the variance of the difference in lack-of-
fit. Since our models are strictly non nested, Rivers and Vuong showed that the asymptotic
distribution of the Tn statistic is standard normal. The selection procedure involves com-
paring the sample value of Tn with critical values of the standard normal distribution9. In
the empirical section, we will present evidence based on these different statistical tests.
9 If α denotes the desired size of the test and tα/2 the value of the inverse standard normal distributionevaluated at 1− α/2. If Tn < tα/2 we reject H0 in favor of H1 ; if Tn > tα/2 we reject H0 in favor of H2.Otherwise, we do not reject H0.
21
5 Econometric Estimation and Test Results
5.1 Data and Variables
Our data were collected by the company SECODIP (Société d’Étude de la Consom-
mation, Distribution et Publicité) that conducts surveys about households’ consumption
in France. We have access to a representative survey for the years 1998, 1999, and 2000.
These data contain information on a panel of nearly 11000 French households and on
their purchases of mostly food products. This survey provides a description of the main
characteristics of the goods and records over the whole year the quantity bought, the
price, the date of purchase and the store where it is purchased. In particular, this survey
contains information on all bottled water purchased by these French households during
the three years of study. We consider purchases of the seven most important retailers
which represent 70.7% of the total purchases of the sample. We take into account the
most important brands, that is five national brands of mineral water, one national brand
of spring water, one retailer private label brand of mineral water and one retailer private
label spring water. The purchases of these eight brands represent 71.3% of the purchases
of the seven retailers. The national brands are produced by three different manufacturers :
Danone, Nestlé and Castel. This survey presents the advantage of allowing to compute
market shares that are representative of the national French market thanks to a weigh-
ting procedure of the available household panel. Then, the market shares are defined by
a weighted sum of the purchases of each brand during each month of the three years
considered divided by the total market size of the respective month. The market share
of the outside good is defined as the difference between the total size of the market and
the shares of the inside goods. We consider all other non-alcoholic refreshing drinks as
the outside good. Therefore, the market size consists in all non-alcoholic refreshing drinks
such as bottled water (sparkling and flavored water), tea drinks, colas, tonics, fruit drinks,
sodas lime. Our data thus allow to compute this market size across all months of the study.
It is clearly varying across periods and shows that the market for non-alcoholic drinks is
affected by seasons or for example the weather.
22
We consider eight brands sold in seven distributors, which gives more than 50 differen-
tiated products in this national market. The number of products in our study thus varies
between 51 and 54 during the 3 years considered. Considering the monthly market shares
of all of these differentiated products, we get a total of 2041 observations in our sample.
For each of these products, we compute an average price for each month. These prices
are in euros per liter (even if until 2000, the money used was the French Franc). Table 1
presents some first descriptive statistics on some of the main variables used.
Variable Mean Median Std. dev. Min. Max
Market share (all inside goods) 0.005 0.003 0.006 4.10−6 0.048Market share : Mineral Water 0.004 0.003 0.003 10−6 0.048Market share : Spring Water 0.010 0.007 0.010 10−5 0.024Price in C=/liter (all inside goods) 0.298 0.323 0.099 0.096 0.823Price in C=/liter : Mineral Water 0.346 0.343 0.060 0.128 0.678Price in C=/liter : Spring Water 0.169 0.157 0.059 0.096 0.276Mineral water dummy (0/1) 0.73 1 0.44 0 1Market Share of the Outside Good 0.71 0.71 0.04 0.59 0.78
Table 1 : Summary Statistics
We also use data from the French National Institute for Statistics and Economic Studies
(INSEE) on the plastic price, on a wage salary index for France, on oil and diesel prices
and on an index for packaging material cost. Over the time period considered (1998-2000),
the wage salary index always raised while the plastic price index first declined during 1998
and the beginning of 1999 before raising again and reaching the 1998 level at the end
of 2000. Concerning the diesel price index, it shows quite an important volatility with a
first general decline during 1998 before a sharp increase until a new decline at the end
of 2000. Also, the packaging material cost index shows important variations with a sharp
growth in 1998, a decline at the beginning of 1999 and again an important growth until the
end of 2000. Interactions of these prices with the dummies for the type of water (spring
versus mineral) will serve as instrumental variables as they are supposed to affect the
marginal cost of production and distribution of bottled water. Actually, it is likely that
labor cost is not the same for the production of mineral or spring water but it is also
known in this industry that the plastic quality used for mineral or spring water is usually
not the same which is also likely to affect their bottling and packaging costs. Also, the
23
relatively important variations of all these price indices during the period of study suggests
a potentially good identification of our cost equations.
5.2 Demand Results
We estimate the demand model (14) which is the following
ln sjt − ln s0t = δj + γt − αpjt + σg ln sjt|g + ujt
as well as the simple logit demand model (15) using two stage least squares in order
to instrument the endogenous variables pjt and ln sjt|g. Results are in Table 2. F tests
of the first stage regressions show that our instrumental variables are well correlated
with the endogenous variables. Moreover, the Sargan test of overidentification validates
the exclusion of excluded instruments from the main equation. The price coefficient has
the expected sign in both specifications and in the case of the nested logit model, the
coefficients σg actually belongs to the [0, 1] interval as required by the theory. Moreover,
since one can reject that parameters σg are zero, it is clear that the nested logit specification
is preferred to the simple logit one for this market of bottled water.
Variable Simple Multinomial Logit Nested Logit
Price (α) (Std. error) 5.04 (1.35) 4.74 (1.56)Mineral water σg (Std. error) 0.52 (0.18)Spring water σg (Std. error) 0.68 (0.17)Coefficients δj , γt not shown
F test that all δj = 0 (p value) 39.79 (0.000) 0.18 (1.000)Wald test that all γt = 0 (p value) 89.89 (0.0000) 64.50 (0.0034)
First stage F test for full model : Stat. (p value) Stat. (p value)Price pjt 5.42 (0.000) 5.43 (0.000)Mineral water ln sjt|g 161.87 (0.000)Spring water ln sjt|g 171.96 (0.000)First stage F test for excluded instruments :Price pjt 7.99 (0.000) 7.26 (0.000)Mineral water ln sjt|g 584.99 (0.000)Spring water ln sjt|g 620.63 (0.000)Sargan test for overidentification 6.302 (0.1777) 6.766 (0.1488)
Table 2 : Estimation Results of Demand Models
In appendix 7.2, we present the first stage regressions of the main demand equation.
Given the demand estimates, it is interesting to note that we find estimates of unobserved
24
product specific mean utilities δj . Using these parameters estimates, one can look at their
correlation with observed product characteristics using ordinary least squares. This is done
in Table 3 below.
Fixed Effects δj Multinomial Logit Nested Logit
Mineral Water (0/1) (Std. error) -2.78 (0.11) -1.20 (0.09)Minerality (Std. error) 0.81 (0.05) 0.75 (0.04)Manufacturer 1 (Std. error) 6.30 (0.11) 4.21 (0.09)Manufacturer 2 (Std. error) 5.65 (0.11) 3.75 (0.09)Manufacturer 3 (Std. error) -3.66 (0.09) -3.51 (0.07)Constant (Std. error) -2.18 (0.06) -2.23 (0.05)F test (p value) 3145.94 (0.000) 3926.94 (0.000)Table 3 : Regression of fixed effects on the product characteristics
Table 3 shows that the product specific constant mean utility δj is increasing with
the minerality of water and that the identity of the manufacturer of the bottled water
affects this mean utility. This is probably due to image, reputation and advertising of the
manufacturing brands. Remark that if one does not control for the manufacturer identity
this mean utility is larger for mineral water rather than spring water but it is not the case
anymore when one introduces these manufacturer dummy variables.
Finally, once we obtained our structural demand estimates, we can compute price elas-
ticities of demand for our differentiated products. Table 4 presents the different average
elasticities obtained for the simple multinomial logit or the nested logit demand model.
All of them have the expected sign and the magnitude of own-price elasticities are much
larger than that of cross-price elasticities. It is interesting to see that in the unrestricted
specification (nested logit), the average own price elasticities are larger than in the restric-
ted (multinomial logit) model. Also average own price elasticities for mineral water and
spring water are almost proportional to average prices of these segments (nearly twice for
mineral water than for spring water) in the case of the multinomial logit model while it is
not the case anymore in the more flexible nested logit model. Average own-price elasticity
for mineral water is actually larger than for spring water but the ratio is not as large as
the one of their average prices. As expected, the cross-price elasticities are larger within
each segment of product than across segments.
25
Average elasticities Multinomial Logit Nested LogitAll bottle waterOwn-price elasticity -9.84 (3.29) -20.72 (4.99)Cross-price elasticity within group 0.05 (0.05) 0.51 (0.55)Cross-price elasticity across group 0.05 (0.05)Mineral waterOwn-price elasticity -11.40 (2.04) -22.20 (3.67)Cross-price elasticity within group 0.05 (0.05) 0.36 (0.31)Cross-price elasticity across group 0.05 (0.05)Spring waterOwn-price elasticity -5.54 (1.97) -16.08 (5.68)Cross-price elasticity within group 0.05 (0.05) 0.99 (0.81)Cross-price elasticity across group 0.05 (0.04)
Table 4 : Summary of Elasticities Estimates
5.3 Price-Cost Margins and Non Nested Tests
Once one has estimated the demand parameters, we can use the formulas obtained in
section 3 to compute the price cost margins at the retailer and manufacturer levels, or
the total price cost margins, for all products, under the various scenarios considered. We
presented several models that seem worth of consideration with some variants on manu-
facturers or retailers behavior. Among the different models with double marginalization or
two part tariffs, we consider the models described in the following table. Each scenario can
be described according to the assumptions made on the manufacturers behavior (collusive
or Nash), the retailers behavior (collusive or Nash) and the vertical interaction which can
be Stackelberg or Nash under double marginalization or with RPM or not under two part
tariffs contracts :
26
Models Retailers Manufacturers VerticalBehavior Behavior Interaction
Double marginalizationModel 1 Collusion Nash NashModel 2 Collusion Nash StackelbergModel 3 Collusion Collusion NashModel 4 Collusion Collusion StackelbergModel 5 Nash Nash NashModel 6 Nash Nash StackelbergModel 7 Nash Collusion NashModel 8 Nash Collusion StackelbergTwo Part TariffsModel 9 Nash Nash RPM10(w = µ)Model 10 Collusion Collusion RPM (w = µ)Model 11 Nash Nash RPM (p = w + c)Model 12 Collusion Collusion RPM (p = w + c)Model 13 Nash Nash no RPMModel 14 : joint profit maximization Collusion Collusion Collusion
Note that in the case of private labels products, we assume that the retailer is also the
producer which amounts in our models to assume that the behavior for pricing private
labels is equivalent to the one of a manufacturer perfectly colluding with the retailer
for this good. Of course, only one price cost margin is then computed for these private
label goods because it has then no meaning to compute wholesale price and retail price
margins separately. Note also that models where perfectly colluding manufacturers would
use two part tariffs contracts with perfectly colluding retailers (models 10 and 12) are
not equivalent to the joint profit maximization of the industry (model 14) because of the
presence of private label products.
Tables 5 and 6 then present the averages11 of product level price cost margins estimates
under the different models with either the logit demand (Table 5) or the more general nes-
ted logit demand (Table 6). It is worth noting that price cost margins are generally lower
for mineral water than for spring water. As done by Nevo (2001), one could then compare
price cost margins with accounting data to evaluate their empirical validity and also even-
10RPM means resale price maintenance. Vertical contracts are such that the producer is always a Sta-ckelberg leader.11Note that the average price-cost margin at the retailer level plus the average price-cost margin at the
manufacturer level do not sum to the total price cost margin because of the private labels products forwhich no price cost margin at the manufacturer level is computed, the retailer price cost margin beingthen equal to the total price cost margin.
27
tually test which model provides the most realistic result. However, the lack of data both
on retailers or manufacturers margins prevents such analysis. Moreover accounting data
only provide an upper bound for price-cost margins.
Price-Cost Margins (% of retail price p) Mineral Water Spring WaterMean Std. Mean Std.
Double MarginalizationModel 1 Retailers 13.47 3.64 30.24 11.49
Manufacturers 9.38 1.18 29.79 2.59Total 22.04 3.35 45.11 26.01
Model 2 Retailers 13.47 3.64 30.24 11.49Manufacturers 10.11 1.36 32.02 3.31Total 22.70 3.48 46.22 27.22
Model 3 Retailers 13.47 3.64 30.24 11.49Manufacturers 11.56 1.62 36.80 4.49Total 24.01 3.97 48.62 29.77
Model 4 Retailers 13.47 3.64 30.24 11.49Manufacturers 15.70 3.22 50.18 10.46Total 27.80 5.66 55.29 37.39
Model 5 Retailers 9.50 2.21 21.34 7.42Manufacturers 9.38 1.18 29.79 2.59Total 18.07 2.58 36.21 22.35
Model 6 Retailers 9.50 2.21 21.34 7.42Manufacturers 9.61 1.61 31.02 3.19Total 18.27 2.74 36.82 22.95
Model 7 Retailers 9.50 2.21 21.34 7.42Manufacturers 11.56 1.62 36.80 4.49Total 20.05 3.14 39.71 26.00
Model 8 Retailers 9.50 2.21 21.34 7.42Manufacturers 12.67 2.52 38.94 4.29Total 21.32 5.05 41.57 27.62
Two part Tariffs with RPMModel 9 Nash and w = µ 9.21 2.17 21.40 8.00Model 10 Collusion and w = µ 11.14 2.11 24.28 10.22Model 11 Nash and p = w+ c 9.82 2.00 25.60 11.70Model 12 Collusion and p = w + c 11.80 2.09 25.60 11.69Two-part Tariffs without RPMModel 13 Retailers 9.50 2.21 21.34 7.42
Manufacturers 2.42 0.66 8.08 2.56Total 11.73 2.05 25.36 11.62
Joint Profit MaximizationModel 14 13.47 3.63 30.24 11.49Table 5 : Price-Cost Margins by groups for the Multinomial Logit Model
28
Price-Cost Margins (% of retail price p) Mineral Water Spring WaterMean Std. Mean Std.
Double MarginalizationModel 1 Retailers 13.27 2.73 29.36 11.25
Manufacturers 6.36 1.07 14.90 2.02Total 18.93 3.23 34.66 17.67
Model 2 Retailers 13.27 2.73 29.36 11.25Manufacturers 6.71 1.33 15.85 2.41Total 19.23 3.40 34.96 18.11
Model 3 Retailers 13.27 2.73 29.36 11.25Manufacturers 11.05 1.46 18.26 2.81Total 23.09 4.38 36.99 20.08
Model 4 Retailers 13.27 2.73 29.36 11.25Manufacturers 14.56 3.61 28.14 6.63Total 26.21 6.27 41.12 25.47
Model 5 Retailers 5.04 0.94 7.65 2.82Manufacturers 6.36 1.07 14.90 2.02Total 10.69 2.23 12.96 9.67
Model 6 Retailers 5.04 0.94 7.65 2.82Manufacturers 7.32 4.31 17.06 3.94Total 12.20 4.31 27.77 4.27
Model 7 Retailers 5.04 0.94 7.65 2.82Manufacturers 11.05 1.46 18.26 2.81Total 14.86 3.63 15.28 11.84
Model 8 Retailers 5.04 0.94 7.65 2.82Manufacturers 14.44 4.88 20.98 5.33Total 17.87 6.24 16.42 13.49
Two part Tariffs with RPMModel 9 Nash and w = µ 5.03 1.25 7.51 3.23Model 10 Collusion and w = µ 9.54 1.57 6.69 2.17Model 11 Nash and p = w+ c 6.42 1.10 8.82 5.03Model 12 Collusion and p = w + c 10.79 1.72 12.00 5.78Two-part Tariffs without RPMModel 13 Retailers 5.04 0.94 7.65 2.82
Manufacturers 3.19 1.10 3.58 2.84Total 7.91 1.58 11.07 7.35
Joint Profit MaximizationModel 14 12.91 3.58 27.33 13.42
Table 6 : Price-Cost Margins by groups for the Nested Logit Model
After estimating the different price cost margins for the models considered, one can
recover the marginal cost Chjt using equation (17) and then estimate (18). The empirical
results of the estimation of these cost equations are in appendix 7.3. They are useful
mostly in order to test which model fits best the data. We thus performed the non nested
tests presented in 4.3. Tables 7 and 8 present the Rivers and Vuong tests for the logit
29
or nested logit demand models. In both cases, the statistics of test12 show that the best
model appears to be the model 9, that is the case where two part tariffs contracts with
resale price maintenance are used by manufacturers with retailers. The Vuong (1989) tests
based on the maximum likelihood estimation of the cost equations under normality draw
the same inference about the best model (see Tables of results of these tests in appendix
7.4).
Rivers and Vuong Test Statistic Tn=√nbσn³Q2n(Θ̂
2
n)−Q1n(Θ̂1
n)´→ N(0, 1)
 H2
H1 2 3 4 5 6 7 8 9 10 11 12 13 141 2.51 2.60 4.59 -2.64 -2.63 -2.40 -1.94 -2.86 -2.82 -2.86 -2.78 -2.78 -2.712 2.07 4.19 -3.20 -3.20 -3.05 -2.72 -3.35 -3.32 -3.35 -3.30 -3.30 -3.263 4.29 -3.35 -3.35 -3.18 -3.11 -3.53 -3.50 -3.53 -3.47 -3.47 -3.424 -5.16 -5.16 -5.04 -4.98 -5.29 -5.27 -5.28 -5.24 -5.24 -5.205 0.57 9.27 3.36 -8.42 -7.41 -8.27 -6.45 -6.96 -1.946 6.11 3.52 -9.80 -8.30 -9.64 -7.30 -6.48 -2.587 2.14 -9.63 -9.17 -9.59 -9.23 -9.31 -8.128 -4.40 -4.21 -4.36 -4.06 -3.99 -3.659 10.02 6.25 10.35 9.96 9.5310 -8.11 6.00 5.26 7.5811 10.61 8.82 9.4512 0.74 7.4213 6.79
Table 7 : Results of the Rivers and Vuong Test for the Multinomial Logit Model
Rivers and Vuong Test Statistic : Tn=√nbσn³Q2n(Θ̂
2
n)−Q1n(Θ̂1
n)´→ N(0, 1)
 H2
H1 2 3 4 5 6 7 8 9 10 11 12 13 141 5.90 6.22 5.99 -9.85 -9.51 -9.78 -9.14 -10.25 -9.94 -10.10 -10.02 -9.71 -9.982 5.59 5.77 -9.82 -9.50 -9.75 -9.23 -10.24 -9.98 -10.09 -10.01 -9.74 -9.833 5.32 -8.82 -8.58 -8.75 -8.36 -9.22 -9.04 -9.09 -9.02 -8.81 -8.554 -7.62 -7.49 -7.55 -7.36 -7.87 -7.79 -7.80 -7.74 -7.64 -7.235 5.73 6.91 6.62 -11.41 -8.26 -10.65 -9.01 -4.70 8.856 -0.22 4.36 -10.58 -8.59 -9.70 -8.79 -6.61 7.637 5.10 -10.51 -8.59 -9.48 -9.76 -6.51 8.628 -9.39 -8.32 -8.65 -8.39 -6.84 5.719 3.50 9.54 10.69 13.16 10.0310 2.58 6.26 8.33 9.2511 6.83 11.16 9.6012 3.43 9.5313 8.62
12Recall that for a 5% size of the test, we reject H0 in favor of H2 if Tn is lower than the critical value-1.64 and that we reject H0 in favor of H1 if Tn is higher than the critical value 1.64.
30
Table 8 : Results of the Rivers and Vuong Test for the Nested Logit Model
Finally, the non rejected model tells that manufacturers use two part tariffs with
retailers and moreover (as predicted by the theory) that they use resale price maintenance
in their contracting relationships although it is in principle not legal in France. For this
model, the estimated total price cost margin (price minus marginal cost of production and
distribution), are relatively low with an average of 5.03% for the mineral water and 7.51%
for spring water. These figures are lower than the rough accounting estimates that one can
get from aggregate data (see section 2). Also, for our best model, we can look at the average
price-cost margins for national brands products versus private labels products. In the case
of mineral water, the average price-cost margins for national brands and private labels
are not statistically different and about the same with an average of 4.91% for national
brands and of 6.77% for private labels. However, in the case of natural spring water, it
appears that price-cost margins for national brands are larger than for private labels with
an average of 10.87% instead of 4.89%. As Nevo (2001) remarks these accounting margins
only provide an upper bound of the true values. Moreover, these accounting estimates do
not take into account the marginal cost of distribution while our structural estimates do.
These empirical results seem then quite realistic and consistent with the bounds provided
by accounting data.
6 Conclusion
We presented how to test across different hypothesis on the strategic relationships bet-
ween manufacturers and retailers in the supermarket industry and in particular how one
can test whether manufacturers use two part tariffs contracts with retailers. We consider
several alternative models of competition between manufacturers and retailers on a diffe-
rentiated product market and test between these alternatives. We consider in particular
two types of non linear pricing relationships, one where resale price maintenance is used
with two part tariffs and one where no resale price maintenance is allowed in two part
tariffs. The method is based on estimates of demand parameters that allow to recover
31
price-cost margins at the manufacturer and retailer levels. We then test between the dif-
ferent models using exogenous variables that are supposed to shift the marginal cost of
production and distribution. We apply this methodology to study the market for retailing
bottled water in France. Our empirical evidence allows to conclude that manufacturers
and retailers use non linear pricing contracts and in particular two part tariffs contracts
with resale price maintenance.
Although, we present the first empirical estimation of a structural model taking into
account explicitly two part tariffs contracts between manufacturers and retailers, this work
calls for further developments and studies about competition under non linear pricing in
the supermarket industry. In particular, further studies where assumptions of constant
marginal cost of production and distribution or more general contractual forms would be
allowed are needed. Also, it is clear that more empirical work on other markets will be
useful for a better understanding of vertical relationships in the retailing industry. Finally
taking into account the endogenous market structure (Rey and Vergé, 2004, started inves-
tigating some theoretical aspects of it) is also an objective that theoretical and empirical
research will have to tackle.
7 Appendix
7.1 Detailed Proof of the Manufacturers Profit Expression under TwoPart Tariffs
We use the theoretical results due to Rey and Vergé (2004) applied to our context
with F firms and R retailers. The participation constraint being binding, we have for all
rPs∈Sr
[M(ps −ws − cs)ss(p)− Fs] = 0 which implies that
Xs∈Sr
Fs =Xs∈Sr
M(ps −ws − cs)ss(p)
and thus
Xj∈Ff
Fj +Xj 6∈Ff
Fj =X
j=1,.,J
Fj =X
r=1,.,R
Xs∈Sr
Fs
=X
r=1,.,R
Xs∈Sr
M(ps −ws − cs)ss(p) =X
j=1,.,J
M(pj −wj − cj)sj(p)
32
so that Xj∈Ff
Fj =X
j=1,..,J
M(pj −wj − cj)sj(p)−Xj 6∈Ff
Fj
Then, the firm f profits are
Πf =Xk∈Ff
M(wk − µk)sk(p) +Xk∈Ff
Fk
=Xk∈Ff
M(wk − µk)sk(p) +X
j=1,..,J
M(pj −wj − cj)sj(p)−Xj 6∈Ff
Fj
Since, producers fix the fixed fees given the ones of other producers, we have that under
resale price maintenance :
max{Fi,pi}i∈Ff
Πf ⇔ max{pi}i∈Ff
Xk∈Ff
(wk − µk)sk(p) +X
j=1,..,J
(pj −wj − cj)sj(p)
⇔ max{pi}i∈Ff
Xk∈Ff
(pk − µk)sk(p) +Xk 6∈Ff
(pk −wk − ck)sk(p)
and with no resale price maintenance
max{Fi,wi}i∈Ff
Πf ⇔ max{wi}i∈Ff
Xk∈Ff
(wk − µk)sk(p) +X
j=1,..,J
(pj −wj − cj)sj(p)
⇔ max{wi}i∈Ff
Xk∈Ff
(pk − µk)sk(p) +Xk 6∈Ff
(pk −wk − ck)sk(p)
Then the first order conditions of the different two part tariffs models can be derived very
simply.
33
7.2 Details on regressions for demand estimates
First Stage Results Dependent VariableMultinomial Logit Nested Logit
Instrumental Variables Price pjt Price pjt ln sjt|g (Spring) ln sjt|g (Mineral)w1t ×1(j∈Mineral water) 0.051 (0.028) 0.053 (0.031) 0.019 (0.016) 0.000 (0.024)w1t × 1(j∈Spring water) 0.045 (0.027) 0.049 (0.030) -0.019 (0.016) 0.008 (0.023)w2t ×1(j∈Mineral water) 0.020 (0.014) 0.028 (0.015) 0.013 (0.008) -0.007 (0.012)w2t ×1(j∈Spring water) -0.001 (0.009) 0.004 (0.012) -0.008 (0.006) 0.004 (0.009)w3t ×1(j∈Mineral water) -0.002 (0.009) -0.003 (0.010) -0.008 (0.005) 0.002 (0.008)w3t × 1(j∈Spring water) -0.007 (0.008) 0.008 (0.009) 0.008 (0.005) -0.005 (0.007)w4t ×1(j∈Mineral water) -0.009 (0.006) -0.010 (0.007) -0.002 (0.004) 0.005 (0.005)w4t × 1(j∈Spring water) 0.002 (0.005) 0.002 (0.006) 0.001 (0.003) -0.000 (0.004)w5t ×1(j∈Mineral water) 0.006 (0.009) 0.008 (0.010) 0.002 (0.005) -0.007 (0.007)w5t × 1(j∈Spring water) -0.005 (0.007) -0.006 (0.007) 0.000 (0.004) -0.001 (0.006)ξgjt × 1(j∈Mineral water) -0.068 (0.032) 1.057 (0.017) 0.002 (0.025)ξgjt × 1(j∈Spring water) -0.019 (0.037) -1.058 (0.020) 0.877 (0.028)Estimates of δ0j, γ0t not shownF test that all δ0j = 0 163.18 153.65 31.89 92.00(p value) (0.000) (0.000) (0.000) (0.000)F test that all γ0t = 0 2.81 2.88 0.62 0.82(p value) (0.000) (0.000) (0.954) (0.756)
Table : First Stage Results for the Demand Estimates
The first stage regressions of the two stage least squares estimation of the demand
models also include some period specific effects (denoted γ0t) and some product specific
effects (denoted δ0j). Moreover, excluded instrumental variables are variables crossing the
mineral water and spring water dummy variables with the wage salary, an oil price index,
a diesel price index, a packaging material cost index and the plastic price. In the case of
the nested logit model estimation, one needs more instrumental variables and we thus use
the variable denoted ξgjt obtained from (16) that is reported in the following table (all the
estimated πj are not reported).
ln sjt|g Coefficient Std.
ln sjt−1|g 0.469 (0.020)Constant -1.992 (0.760)F test that all πj = 0 10.97 (0.000)F test (p value) 549.10 (0.000)
Table : Auxiliary regression to define ξgjt
34
7.3 Estimates of Cost Equations
Here, we present the empirical results of the estimation of the cost equation (18) for
h = 1, ..., 14 that is
lnChjt = ωhj +Wjtγg + ln η
hjt
where variables Wjt include time dummies δt, wages, oil, diesel, packaging material and
plastic price variables interacted with the dummy variable for spring water (SW ) and
mineral water (MW ).
Coefficients (Std. err.)lnCh
jt salary×SW salary×MW plastic×SW plastic×MW packaging×SW packaging×MW
Model 1 -0.316 (0.032) -0.109 (0.025) -0.074 (0.014) -0.039 (0.012) 0.071 (0.011) 0.020 (0.009)Model 2 -0.504 (0.042) -0.147 (0.032) -0.127 (0.018) -0.054 (0.015) 0.098 (0.014) 0.030 (0.012)Model 3 -0.318 (0.041) -0.110 (0.030) -0.062 (0.018) -0.037 (0.014) 0.063 (0.013) 0.026 (0.011)Model 4 0.040 (0.058) -0.175 (0.040) 0.090 (0.024) -0.055 (0.019) 0.000 (0.017) 0.039 (0.014)Model 5 -0.021 (0.015) -0.008 (0.012) 0.001 (0.007) -0.009 (0.006) 0.002 (0.005) 0.006 (0.004)Model 6 -0.036 (0.015) -0.009 (0.012) -0.001 (0.007) -0.010 (0.006) 0.005 (0.005) 0.006 (0.004)Model 7 -0.107 (0.018) -0.042 (0.014) -0.020 (0.008) -0.020 (0.007) 0.020 (0.006) 0.013 (0.005)Model 8 -0.165 (0.021) -0.057 (0.017) -0.035 (0.009) -0.024 (0.008) 0.034 (0.007) 0.014 (0.006)Model 9 0.002 (0.013) 0.008 (0.010) 0.005 (0.006) -0.004 (0.005) -0.002 (0.004) 0.003 (0.004)Model 10 -0.019 (0.014) -0.008 (0.011) 0.001 (0.006) -0.008 (0.005) 0.003 (0.005) 0.006 (0.004)Model 11 -0.007 (0.013) -0.002 (0.011) 0.003 (0.006) -0.006 (0.005) -0.000 (0.005) 0.004 (0.004)Model 12 -0.076 (0.014) -0.040 (0.011) -0.014 (0.006) -0.017 (0.005) 0.014 (0.005) 0.012 (0.004)Model 13 -0.027 (0.014) -0.011 (0.011) -0.000 (0.006) -0.007 (0.005) 0.004 (0.005) 0.006 (0.004)Model 14 -0.133 (0.015) -0.066 (0.012) -0.027 (0.007) -0.024 (0.006) 0.025 (0.005) 0.016 (0.004)
Table : Cost Equations for the Multinomial Logit ModelCoefficients (Std. err.) All δt = 0 All ωgj = 0
lnChjt diesel×SW diesel×MW oil×SW oil×MW F test (p val.) F test (p val.)
Model 1 -0.013 (0.007) 0.006 (0.006) 0.040 (0.009) 0.007 (0.008) 8.01 (0.000) 274.39 (0.000)Model 2 -0.003 (0.009) 0.007 (0.007) 0.043 (0.012) 0.008 (0.010) 5.80 (0.000) 189.82 (0.000)Model 3 -0.027 (0.008) 0.004 (0.006) 0.058 (0.011) 0.008 (0.009) 3.35 (0.000) 250.34 (0.000)Model 4 -0.061 (0.012) -0.004 (0.009) 0.058 (0.015) 0.024 (0.013) 2.08 (0.001) 218.02 (0.000)Model 5 -0.005 (0.003) -0.000 (0.003) 0.011 (0.005) 0.003 (0.004) 1.67 (0.011) 783.26 (0.000)Model 6 -0.005 (0.003) -0.000 (0.003) 0.012 (0.005) 0.003 (0.004) 1.72 (0.008) 796.10 (0.000)Model 7 -0.006 (0.004) 0.001 (0.003) 0.018 (0.005) 0.004 (0.005) 2.64 (0.000) 729.80 (0.000)Model 8 -0.009 (0.004) 0.003 (0.004) 0.024 (0.006) 0.004 (0.005) 3.47 (0.000) 599.76 (0.000)Model 9 -0.004 (0.003) -0.001 (0.003) 0.007 (0.004) 0.002 (0.003) 1.29 (0.133) 560.97 (0.000)Model 10 -0.005 (0.003) -0.000(0.002) 0.010 (0.004) 0.002 (0.004) 1.16 (0.251) 535.56 (0.000)Model 11 -0.004 (0.003) -0.001 (0.002) 0.008 (0.004) 0.002 (0.003) 1.47 (0.045) 557.61 (0.000)Model 12 -0.005 (0.003) 0.001 (0.002) 0.014 (0.004) 0.004 (0.004) 3.43 (0.000) 550.13 (0.000)Model 13 -0.005 (0.003) -0.001 (0.002) 0.012 (0.004) 0.004 (0.003) 1.89 (0.002) 562.73 (0.000)Model 14 -0.005 (0.003) 0.001 (0.003) 0.020 (0.004) 0.007 (0.004) 7.25 (0.000) 519.31 (0.000)
Table (continued) : Cost Equations for the Multinomial Logit Model
35
Coefficients (Std. err.)lnCh
jt salary×SW salary×MW plastic×SW plastic×MW packaging×SW packaging×MW
Model 1 -0.172 (0.023) -0.006 (0.018) -0.010 (0.010) 0.000 (0.008) -0.004 (0.010) -0.025 (0.009)Model 2 -0.206 (0.025) -0.005 (0.020) -0.017 (0.011) -0.000 (0.008) -0.003 (0.011) -0.028 (0.010)Model 3 -0.257 (0.028) -0.010 (0.022) -0.029 (0.012) -0.000 (0.009) -0.003 (0.012) -0.035 (0.010)Model 4 0.027 (0.036) -0.046 (0.025) 0.062 (0.014) -0.003 (0.011) -0.037 (0.014) -0.024 (0.012)Model 5 -0.007 (0.013) 0.004 (0.010) 0.011 (0.006) 0.000 (0.004) -0.003 (0.006) 0.000 (0.005)Model 6 -0.006 (0.019) 0.004 (0.015) 0.017 (0.008) -0.003 (0.006) -0.010 (0.0081) 0.002 (0.007)Model 7 -0.015 (0.014) -0.002 (0.011) 0.010 (0.006) 0.001 (0.005) -0.005 (0.006) -0.002 (0.005)Model 8 -0.018 (0.016) 0.012 (0.013) 0.011 (0.007) 0.006 (0.005) -0.005 (0.007) -0.005 (0.006)Model 9 0.000 (0.012) 0.005 (0.010) 0.008 (0.005) -0.001 (0.004) -0.001 (0.005) 0.002 (0.005)Model 10 -0.004 (0.014) -0.006 (0.011) 0.005 (0.006) -0.004 (0.005) 0.001 (0.006) 0.006 (0.005)Model 11 -0.003 (0.012) 0.004 (0.010) 0.009 (0.005) 0.000 (0.004) -0.003 (0.005) 0.001 (0.005)Model 12 -0.031 (0.015) -0.007 (0.011) 0.006 (0.006) -0.001 (0.005) -0.005 (0.006) -0.004 (0.006)Model 13 -0.008 (0.013) 0.002 (0.011) 0.010 (0.006) 0.000 (0.004) -0.000 (0.006) 0.001 (0.005)Model 14 -0.097(0.018) -0.008 (0.014) 0.001 (0.007) 0.003 (0.006) -0.006 (0.007) -0.019 (0.007)
Table : Cost Equations for the Nested Logit ModelCoefficients (Std. err.) All δt = 0 All ωgj = 0
lnChjt diesel×SW diesel×MW oil×SW oil×MW F test (p val.) F test (p val.)
Model 1 0.013 (0.005) 0.014 (0.004) 0.000 (0.007) -0.010 (0.006) 5.38 (0.000) 300.63 (0.000)Model 2 0.017 (0.005) 0.016 (0.005) -0.001 (0.008) -0.013 (0.006) 5.48 (0.000) 274.99 (0.000)Model 3 0.024 (0.006) 0.020 (0.005) -0.005 (0.008) -0.016 (0.007) 6.01 (0.000) 244.69 (0.000)Model 4 -0.010 (0.008) 0.015 (0.006) 0.007 (0.010) -0.006 (0.008) 3.13 (0.000) 233.37 (0.000)Model 5 -0.002 (0.003) -0.001 (0.002) 0.005 (0.004) 0.003 (0.003) 1.09 (0.345) 495.72 (0.000)Model 6 -0.001 (0.004) -0.001 (0.003) 0.006 (0.006) 0.002 (0.005) 1.02 (0.435) 276.84(0.000)Model 7 -0.000 (0.003) 0.001 (0.003) 0.004 (0.004) 0.002 (0.004) 1.21 (0.200) 473.69 (0.000)Model 8 0.001 (0.004) -0.000 (0.003) 0.002 (0.005) 0.002 (0.004) 1.23 (0.190) 383.82 (0.000)Model 9 -0.003 (0.003) 0.002 (0.002) 0.006 (0.004) 0.003 (0.003) 1.08 (0.356) 473.31 (0.000)Model 10 -0.005 (0.003) -0.002 (0.003) 0.008 (0.004) 0.003 (0.004) 0.84 (0.707) 236.63 (0.000)Model 11 -0.002 (0.003) -0.001 (0.002) 0.005 (0.004) 0.003 (0.003) 1.13 (0.287) 490.63 (0.000)Model 12 -0.000 (0.003) 0.002 (0.003) 0.002 (0.003) 0.006 (0.004) 1.97 (0.002) 298.09 (0.000)Model 13 -0.004 (0.003) -0.001 (0.003) 0.007 (0.004) 0.003 (0.003) 1.18 (0.238) 452.71 (0.000)Model 14 0.006 (0.004) 0.009 (0.003) 0.006 (0.005) -0.006 (0.005) 6.33 (0.000) 350.49 (0.000)
Table (continued) : Cost Equations for the Nested Logit Model
36
7.4 Additional Non Nested Tests
Vuong (1989) Test Statistic H2
H1 2 3 4 5 6 7 8 9 10 11 12 13 141 3.74 3.34 8.01 -4.96 -4.93 -4.10 -2.78 -6.06 -5.68 -6.02 -5.62 -5.64 -5.3962 1.70 5.77 -7.10 -7.16 -6.32 -4.93 -8.27 -7.87 -8.23 -7.82 -7.84 -7.673 6.46 -7.01 -7.08 -6.17 -5.74 -8.25 -7.83 -8.21 -7.77 -7.78 -7.524 -12.93 -13.16 -11.84 -11.38 -14.57 -14.02 -14.51 -13.89 -13.90 -13.545 0.56 13.18 4.39 -12.63 -7.32 -12.23 -7.81 -9.29 -2.036 7.77 4.92 -11.63 -7.06 -11.29 -7.07 -6.67 -2.587 2.51 -15.63 -12.10 -15.53 -13.27 -14.69 -11.998 -6.65 -5.85 -6.58 -5.77 -5.67 -5.069 8.80 6.87 16.42 12.47 12.1910 -7.07 2.36 1.60 6.4111 14.26 12.21 12.0012 0.46 7.3013 7.23
Table : Results of the Vuong Test for the Multinomial Logit Model
Vuong (1989) Test Statistic H2
H1 2 3 4 5 6 7 8 9 10 11 12 13 141 9.17 7.68 7.98 -15.49 -1.06 -14.98 -3.95 -15.34 -10.75 -15.65 -11.21 -13.57 -15.352 6.64 6.88 -15.46 -1.39 -14.96 -4.62 -15.56 -11.40 -15.78 -11.70 -13.98 -14.653 5.46 -14.46 -1.79 -13.96 -5.32 -14.90 -11.47 -14.98 -11.56 -13.47 -12.574 -14.94 -2.44 -14.55 -6.55 -15.23 -12.15 -15.30 -12.35 -14.54 -12.105 1.58 12.88 2.09 -10.29 3.61 -11.47 5.49 0.47 12.516 -1.35 -0.64 -2.00 1.22 -1.91 -1.13 -1.54 -0.277 1.56 -11.68 1.22 -12.97 2.49 -2.15 11.048 -3.06 -1.21 -2.85 -1.02 -1.96 0.989 9.65 6.67 12.09 5.49 13.2510 -8.13 2.40 -2.64 6.9011 10.86 4.56 13.5512 -3.63 7.4013 9.59
Table : Results of the Vuong Test for the Nested Logit Model
37
References :
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